Alain Brobecker - 99dec24 Below is a small program (i was lucky it was a finite problem) which looks for k-Multicyclic numbers, ie numbers with decimal expression d(0)...d(n) such that 0<=d(i)<=9, d(0)<>0, d(n)<>0 and d(0)...d(n)*k=d(n)d(0)...d(n-1). There is an infinity of k-Multicyclic numbers: 142857 x 5 = 142857142857 x 5 = 142857142857142857 x 5 = ... etc ---------- 8< ---------- 8< ---------- 8< ---------- REM >kMulti.bas - 24dec1999 REM Alain Brobecker FOR Multiplier%=2 TO 9 MaxLeftDigit%=INT(10/Multiplier%) IF MaxLeftDigit%=10/Multiplier% THEN MaxLeftDigit%-=1 MaxCarry%=INT(Multiplier%*9/10) FOR LeftDigit%=0 TO MaxLeftDigit% FOR LeftCarry%=0 TO MaxCarry% RightDigit%=Multiplier%*LeftDigit%+LeftCarry% IF RightDigit%<=9 THEN Carry%=0 Digit%=RightDigit% Number$=STR$Digit% WHILE (Carry%<>LeftCarry%) OR (Digit%<>LeftDigit%) Temp%=Multiplier%*Digit%+Carry% Carry%=INT(Temp%/10) Digit%=Temp%-10*Carry% Number$=STR$Digit%+Number$ ENDWHILE ENDIF PRINT Number$;" ";Multiplier%;" * " NEXT LeftCarry% NEXT LeftDigit% NEXT Multiplier% ---------- 8< ---------- 8< ---------- 8< ---------- 2-MultiCyclic numbers 052631578947368421 105263157894736842 157894736842105263 210526315789473684 263157894736842105 315789473684210526 368421052631578947 421052631578947368 473684210526315789 3-MultiCyclic numbers 0344827586206896551724137931 0689655172413793103448275862 1034482758620689655172413793 1379310344827586206896551724 1724137931034482758620689655 2068965517241379310344827586 2413793103448275862068965517 2758620689655172413793103448 3103448275862068965517241379 4-MultiCyclic numbers 025641 051282 076923 102564 128205 153846 179487 205128 230769 5-MultiCyclic numbers 020408163265306122448979591836734693877551 040816326530612244897959183673469387755102 061224489795918367346938775510204081632653 081632653061224489795918367346938775510204 102040816326530612244897959183673469387755 122448979591836734693877551020408163265306 142857 163265306122448979591836734693877551020408 183673469387755102040816326530612244897959 6-MultiCyclic numbers 0169491525423728813559322033898305084745762711864406779661 0338983050847457627118644067796610169491525423728813559322 0508474576271186440677966101694915254237288135593220338983 0677966101694915254237288135593220338983050847457627118644 0847457627118644067796610169491525423728813559322033898305 1016949152542372881355932203389830508474576271186440677966 1186440677966101694915254237288135593220338983050847457627 1355932203389830508474576271186440677966101694915254237288 1525423728813559322033898305084745762711864406779661016949 7-MultiCyclic numbers 0144927536231884057971 0289855072463768115942 0434782608695652173913 0579710144927536231884 0724637681159420289855 0869565217391304347826 1014492753623188405797 1159420289855072463768 1304347826086956521739 8-MultiCyclic numbers 0126582278481 0253164556962 0379746835443 0506329113924 0632911392405 0759493670886 0886075949367 1012658227848 1139240506329 9-MultiCyclic numbers 01123595505617977528089887640449438202247191 02247191011235955056179775280898876404494382 03370786516853932584269662921348314606741573 04494382022471910112359550561797752808988764 05617977528089887640449438202247191011235955 06741573033707865168539325842696629213483146 07865168539325842696629213483146067415730337 08988764044943820224719101123595505617977528 10112359550561797752808988764044943820224719